1. © Math Workshop at EDC, Inc., 2005 Learning by Doing, for Teachers A project of With support from www2.edc.org/mathworkshop From a new comprehensive K-5 NSF program by EDC and Harcourt School Publishers Algebraic ideas from arithmetic NCSM — Anaheim, 2005

2. © Math Workshop at EDC, Inc., 2005 To serve the teacher… Connect directly with their practice Capture time they already have If it’s curriculum, it must Serve the children Capture the adult’s curiosity Meet the needs of the job of teaching tests Allow easy start without independent PD To work, PD must improve life for the teacher. It must start where teachers are and acknowledge what they do know as well as what they might not know.

3. © Math Workshop at EDC, Inc., 2005 …what could be less sexy than memorizing 4th grade multiplication facts?

4. © Math Workshop at EDC, Inc., 2005 Just the facts Start by knowing 4  4, 5  5, 6  6, 7  7, … Have most others and easily work out what they don’t have memorized. Goal now is to consolidate!

5. © Math Workshop at EDC, Inc., 2005 What helps kids memorize multiplication facts? Something memorable!

6. © Math Workshop at EDC, Inc., 2005 Surprise What is 6  6?

7. © Math Workshop at EDC, Inc., 2005 Surprise What is 6  6? What is 5  7? 36 35

8. © Math Workshop at EDC, Inc., 2005 Surprise What is 7  7?

9. © Math Workshop at EDC, Inc., 2005 Surprise What is 7  7? What is 6  8? 49 48

10. © Math Workshop at EDC, Inc., 2005 Surprise What is 8  8?

11. © Math Workshop at EDC, Inc., 2005 Surprise What is 8  8? What is 7  9? 64 63

12. © Math Workshop at EDC, Inc., 2005 Surprise What is 9  9?

13. © Math Workshop at EDC, Inc., 2005 Surprise What is 9  9? What is 8  10? 81 80

14. © Math Workshop at EDC, Inc., 2005 Is this always true?

15. © Math Workshop at EDC, Inc., 2005 Is this always true? always one more than Is this number times itself

16. © Math Workshop at EDC, Inc., 2005 Is this always true? always one more than Is this number times itself the product of these two numbers?

17. © Math Workshop at EDC, Inc., 2005 But why does it work?!

18. © Math Workshop at EDC, Inc., 2005 One way to look at it 5  5

19. © Math Workshop at EDC, Inc., 2005 One way to look at it 5  4 Removing a column leaves

20. © Math Workshop at EDC, Inc., 2005 One way to look at it 6  4 Replacing as a row leaves with one left over.

21. © Math Workshop at EDC, Inc., 2005 One way to look at it 6  4 Removing the leftover leaves showing that it is one less than 5  5.

22. © Math Workshop at EDC, Inc., 2005 A second look Don’t bother counting! A square array.

23. © Math Workshop at EDC, Inc., 2005 A second look Removing a column leaves it narrower by 1.

24. © Math Workshop at EDC, Inc., 2005 A second look Replacing as a row leaves it narrower by 1 and taller by 1 (with 1 left over).

25. © Math Workshop at EDC, Inc., 2005 A second look Removing the leftover shows that the new array contains one less dot than the square.

26. © Math Workshop at EDC, Inc., 2005 What’s the gain? An aid for remembering 6  8 or 7  9  7  7 = 49  6  8 = 48  (6  8) = (7  7) - 1 Direct benefit!

27. © Math Workshop at EDC, Inc., 2005 What’s the gain? An aid for remembering 6  8 or 7  9  7  7 = 49  6  8 = 48  (6  8) = (7  7) – 1 A practical tool for (some) calculations A hint at a BIG IDEA lurking Investment in the future!

28. © Math Workshop at EDC, Inc., 2005 Further Investigation In the process of taking this idea further, the children get more multiplication practice. Is there a pattern that lets us use 7  7…

29. © Math Workshop at EDC, Inc., 2005 Further Investigation In the process of taking this idea further, the children get more multiplication practice. Is there a pattern that lets us use 7  7 to derive 5  9?

30. © Math Workshop at EDC, Inc., 2005 Experiment a moment Find a pattern that shows how 7  7 relates to 5  9…

31. © Math Workshop at EDC, Inc., 2005 Experiment a moment …or how 8  8 relates to 6  10…

32. © Math Workshop at EDC, Inc., 2005 Experiment a moment …or how 9  9 relates to 7  11…

33. © Math Workshop at EDC, Inc., 2005 (7 – 1) (7 + 1) = 7  7 – 1 n n – 1 n + 1 Or use 9 as an example (9 – 1)  (9 + 1) = 9  9 – 1 8  10 = 81 – 1

34. © Math Workshop at EDC, Inc., 2005 n n – 2 n + 2 (7 – 2) (7 + 2) = 7  7 – 4 Or use 8 as an example (8 – 2)  (8 + 2) = 8  8 – 4 6  10 = 64 – 4

35. © Math Workshop at EDC, Inc., 2005 n n – 3 n + 3 (7 – 3) (7 + 3) = 7  7 – 9 Or use 10 as an example (10 – 3)  (10 + 3) = 10  10 – 9 7  13 = 100 – 9

36. © Math Workshop at EDC, Inc., 2005 Where does this lead?

37. © Math Workshop at EDC, Inc., 2005 Where does this lead? To do… 53  47

38. © Math Workshop at EDC, Inc., 2005 Where does this lead? To do……I think… 533 more than 50  47

39. © Math Workshop at EDC, Inc., 2005 Where does this lead? To do……I think… 533 more than 50  473 less than 50 50  50 (well, 5  5 and …) …2500 Minus 3  3 – 9

40. © Math Workshop at EDC, Inc., 2005 Where does this lead? To do……I think… 533 more than 50  473 less than 50 50  50 (well, 5  5 and …) …2500 Minus 3  3 – 9 2491

41. © Math Workshop at EDC, Inc., 2005 Why does it work? 473 50 53

42. © Math Workshop at EDC, Inc., 2005 www2.edc.org/mathworkshop Thanks! E. Paul Goldenberg pgoldenberg@edc.org Contact Information

43. © Math Workshop at EDC, Inc., 2005 www2.edc.org/mathworkshop Bye! Thanks! E. Paul Goldenberg pgoldenberg@edc.org Contact Information